Comparative analysis of the difference and probabilistic computational models to study partial differential equation of the elliptical type
Authors: Kryukov S.A.  
Published in issue: #5(82)/2023  
DOI: 10.18698/2541800920235894  
Category: Mathematics  Chapter: Computational Mathematics 

Keywords: temperature field simulation, elliptic equation, difference scheme, particle random walk, Dirichlet boundary conditions, probabilistic method, difference method 

Published: 19.05.2023 
The paper presents a comparative analysis of the difference and probabilistic computational models to study the heat transfer equation with the boundary conditions of the I kind. The first model was implemented by an explicit fivepoint scheme, and the second — by the random walk algorithm. The proposed approaches were confirmed by simulating the temperature field in a thin rectangular plate exposed to external radiation sources. During a series of experiments, it was found that the probabilistic method for determining the temperature value at a single point made it possible to obtain an approximate result from 2 to 200 times faster, depending on the number of irradiated particles and requird less memory compared to the difference method. It was established that, if the point under study was located close to one of the boundaries, the difference between results obtained during the algorithms operation was on average 1°C. And if its location was in the middle of the plate, the differences reached 3°C. The probabilistic model in most cases would lead to the incorrect result with a small number of the grid nodes. Meanwhile, the difference model demonstrates values that are more accurate and determines the thermal field completely.
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